Is there a difference between the expressions 元宵节 and 灯节, or are they exactly the same?
元：上元 (農曆正月十五，the 15th of the 1st lunar month)
宵：夜晚 (night, evening)
元宵：the evening of the 15th of the 1st lunar month
Lantern Festival: 燈節
The Lantern Festival falls on the 15th day of the 1st lunar month, usually in February or March in the Gregorian calendar. As early as the Western Han Dynasty (206 BC-AD 25), it had become a festival with great significance. This day's important activity is watching lanterns. Throughout the Han Dynasty (206 BC-AD 220), Buddhism flourished in China. One emperor heard that Buddhist monks would watch sarira, or remains from the cremation of Buddha's body, and light lanterns to worship Buddha on the 15th day of the 1st lunar month, so he ordered to light lanterns in the imperial palace and temples to show respect to Buddha on this day. Later, the Buddhist rite developed into a grand festival among common people and its influence expanded from the Central Plains to the whole of China.
Complementary to @hinen answer.
灯节 , is actually geographical cultural specific.
For ethnic "Han" and to the confusion of other cultural, Lantern festival may refer any of these day in agri-lunar(农历）Calendar
- 农历正月十五 元宵节 (1st month, day 15)
- 农历七月七日 七巧节 (7th month, day 7 )
- 农历八月十五 中秋节 (8th month, day 15)
灯节is an informal term of 元宵节。
元宵 doesn't means 'lantern'. It means 'the night of the 15th day of January', because January(一月) is also called 元月 in ancient Chinese.
宵 means 'night'.
In old tradition, people see lantern shows outside at this day. Thus it is also called 灯节（lantern festival）。